Binomial coefficient structure
The binomial power determines the Pascal row and the coefficient pattern. Factor powers move in opposite directions, and subtraction produces alternating signs
\[\binom{n}{r}=\frac{n!}{r!\left(n-r\right)!}\]
- Match the row index to the power For a non-negative integer n, Pascal row P_n has n+1 entries. Row P_0 is (1), so the row index equals the binomial power rather than the number of entries \(\displaystyle \left(x+y\right)^4\longrightarrow P_4\)
- Add adjacent parents Each interior entry is the sum of the two entries directly above it \(\displaystyle P_4=\left(1,4,6,4,1\right)\)
- Apply alternating signs In (A−B)^n, include the sign in (−B)^r. Odd r gives a negative sign and even r a positive sign before any variable values are substituted \(\displaystyle \left(x-y\right)^3=x^3-3x^2y+3xy^2-y^3\)
- Keep factor powers balanced In the r-indexed term, raise the entire first factor to n−r and the entire second factor to r. Their factor exponents add to n; multiply by the Pascal coefficient as well \(\displaystyle \left(x+y\right)^4:\quad\left(4,0\right),\left(3,1\right),\left(2,2\right),\left(1,3\right),\left(0,4\right)\)
- Include scalar factors The Pascal entry alone is not the final coefficient when either factor contains a numerical multiplier \(\displaystyle T_{r+1}=\binom nr A^{n-r}B^r,\quad r=0,1,\ldots,n\)
Worked example
Complete the coefficient and factor-power data for the expansion below
\[\left(3a-2b\right)^3\]
- \(\displaystyle P_3=\left(1,3,3,1\right)\)The power 3 selects Pascal row P3.
- \(\displaystyle 3^{3}=27,\quad3\times3^{2}\times\left(-2\right)=-54,\quad3\times3\times\left(-2\right)^{2}=36,\quad\left(-2\right)^{3}=-8\)Apply the scalar powers and the alternating signs to each row entry
- \(\displaystyle C=\left(27,-54,36,-8\right)\)These four values form the coefficient row after the scalar factors are included
- \(\displaystyle \left(3a-2b\right)^3=27a^3-54a^2b+36ab^2-8b^3\)The power of a falls while the power of b rises, and each pair sums to 3.
Common mistakes
Using the wrong Pascal row
✕\(\displaystyle \left(x+y\right)^4\longrightarrow P_3=\left(1,3,3,1\right)\)✓\(\displaystyle \left(x+y\right)^4\longrightarrow P_4=\left(1,4,6,4,1\right)\)
The row index is the total binomial power, so power 4 requires P4.Not alternating subtraction signs
✕\(\displaystyle \left(x-y\right)^4=x^4+4x^3y+6x^2y^2+4xy^3+y^4\)✓\(\displaystyle \left(x-y\right)^4=x^4-4x^3y+6x^2y^2-4xy^3+y^4\)
Each term containing an odd power of the subtracted factor has a negative signMoving both factor powers together
✕\(\displaystyle \left(x+y\right)^4:\quad\left(4,0\right),\left(5,1\right),\left(6,2\right),\left(7,3\right),\left(8,4\right)\)✓\(\displaystyle \left(x+y\right)^4:\quad\left(4,0\right),\left(3,1\right),\left(2,2\right),\left(1,3\right),\left(0,4\right)\)
One factor power decreases as the other increases, so every pair must add to 4.Difficulty levels
introIdentify short Pascal rows and their interior coefficients
standardMatch coefficients to factor powers in short expansions
challengeUse longer rows, scalar factors and subtraction signs
examCombine coefficient, sign and factor-power information
Questions about this topic
Why does the row index equal the binomial power?+
Pascal row P_n contains the coefficients for a binomial raised to power n.
How can the factor powers be checked?+
The two powers should move in opposite directions and add to the total binomial power in every term
Which terms are negative in a subtraction expansion?+
Terms with an odd power of the second, subtracted factor are negative; terms with an even power are positive
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 2: Algebraic foundations
In this chapter
Expand binomial and special productsFactorise by common factors and difference of perfect squaresFactorise monic and non-monic quadratic trinomialsFactorise four-term expressions by groupingRecognise equivalent polynomial formsSimplify and operate on algebraic fractionsSimplify and combine algebraic fractionsEvaluate factorial and combination expressionsPascal rows and binomial coefficient structureLocate and evaluate a term in a binomial expansionClassify real numbers and recognise undefined formsWork with sets and interval notationSimplify, compare and operate with surdsRationalise a surd denominator